A Sharp Rate of Convergence for the Empirical Spectral Measure of a Random Unitary Matrix

Abstract

We consider the convergence of the empirical spectral measures of random N × N unitary matrices. We give upper and lower bounds showing that the Kolmogorov distance between the spectral measure and uniform measure on the unit circle is of order log N/N, both in expectation and almost surely. This implies, in particular, that the convergence happens more slowly for Kolmogorov distance than for the L₁-Kantorovich distance. The proof relies on the determinantal structure of the eigenvalue process.